The local linear approximation to the function $g$ at $x=6$ is $y=-3x+4$. What is the value of $g(6)+g'(6)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $-17$ (Choice B) B $-18$ (Choice C) C $-19$ (Choice D) D $-20$
Explanation: The local linear approximation of $g$ at $x=6$ is achieved using the equation of the line tangent to $g$ at $x=6$. In other words, $y=-3x+4$ is the equation of the line tangent to the graph of $g$ at $x=6$. How can we use this to find $g(6)$ and $g'(6)$ ? Since the line is tangent to the graph of $g$ at $x=6$, we know two key facts about it: The line passes through the point $({6},{g(6)})$ The line's slope is ${g'(6)}$ The slope of $y={-3}x+4$ is ${-3}$. The $y$ -value that corresponds to $x={6}$ is $-3({6})+4={-14}$. Now we can find our answer: ${g(6)}+{g'(6)}={-14}+({-3})=-17$